The invention relates broadly to apparatus for filtering air. More particularly, the invention relates to an apparatus for removing dust and fibrous material from the ambient air in a surrounding enclosure or space, especially that of the machine rooms of textile manufacturing factories. The invention especially relates to improvements in filtering apparatus in which a rotating filter drum receives air from the outside and discharges it from the interior of the drum while a suction mechanism removes the trapped or filtered-out material from the circumference of the filtering drum. Depending on the permeability of the filtering medium, which is disposed on the circumference of the filtering drum, the filter either traps only fibers or larger contaminants, such as fiber fragments, coarse dust, and the like, and will consequently permit fine dust to pass through unimpeded. In that case, the fine particles will be carried along with the air after passage through the filtering apparatus and will thus require a second filtering in a subsequent fine filter. Inasmuch as the entire air flow must pass through the filter, this second filter must also have a throughput rate equal to the total air flow, which makes it expensive. Two-stage filtering systems of this type are expensive and require a substantial space for installation. If the first drum filter is dispensed with, i.e., if only a fine dust filter is provided, the coarser contaminants, i.e., fiber remnants and coarse dust, are not separated from the more finely distributed dust which makes it difficult or even impossible to recover the fibrous remnants for other purposes. It is also known in the art to provide as a first filtering system a so-called cyclone separator, for example as described in the German Offenlegungsschrift No. 1 908 442. Such a separator, however, still requires a second filtering stage to remove the fine dust which thus is still required to have dimensions capable of handling the entire air flow rate that passes through the first filtering stage.